Investigating Second Eigenvalue

Investigating Second Eigenvalue

  if (require("PageRank")) {
      library(PageRank)
    }else{
      devtools::install_github("ryangreenup/PageRank")
      library(PageRank)
    }

  library(pacman)
  pacman::p_load(PageRank, devtools, Matrix, igraph, mise, tidyverse, rgl, latex2exp)
#  mise()

Looking at Density

Constants

Define some constants

p    <- seq(from = 0.01, to = 0.99, length.out = 10)
beta <- seq(from = 1, to = 20, length.out = 4)
sz <- seq(from = 100, to = 10, length.out = 5)
input_var <- expand.grid("p" = p, "beta" = beta, "size" = sz)
input_var

Function to Build Graph

random_graph <- function(p, beta, size) {
      g1 <- igraph::erdos.renyi.game(n = size, p)
      A <- igraph::get.adjacency(g1) # Row to column
      A <- Matrix::t(A)

      A_dens <- mean(A)
      T      <- PageRank::power_walk_prob_trans(A, beta = beta)
      tr     <- sum(diag(T))
      e2     <- eigen(T, only.values = TRUE)$values[2] # R orders by descending magnitude
      return(c(abs(e2), mean(A), tr))
}

Return results

Map the function

nc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$p, X$beta, X$size)
  print(i/nrow(input_var))
}

[1] 0.005
[1] 0.01
[1] 0.015
[1] 0.02
[1] 0.025
[1] 0.03
[1] 0.035
[1] 0.04
[1] 0.045
[1] 0.05
[1] 0.055
[1] 0.06
[1] 0.065
[1] 0.07
[1] 0.075
[1] 0.08
[1] 0.085
[1] 0.09
[1] 0.095
[1] 0.1
[1] 0.105
[1] 0.11
[1] 0.115
[1] 0.12
[1] 0.125
[1] 0.13
[1] 0.135
[1] 0.14
[1] 0.145
[1] 0.15
[1] 0.155
[1] 0.16
[1] 0.165
[1] 0.17
[1] 0.175
[1] 0.18
[1] 0.185
[1] 0.19
[1] 0.195
[1] 0.2
[1] 0.205
[1] 0.21
[1] 0.215
[1] 0.22
[1] 0.225
[1] 0.23
[1] 0.235
[1] 0.24
[1] 0.245
[1] 0.25
[1] 0.255
[1] 0.26
[1] 0.265
[1] 0.27
[1] 0.275
[1] 0.28
[1] 0.285
[1] 0.29
[1] 0.295
[1] 0.3
[1] 0.305
[1] 0.31
[1] 0.315
[1] 0.32
[1] 0.325
[1] 0.33
[1] 0.335
[1] 0.34
[1] 0.345
[1] 0.35
[1] 0.355
[1] 0.36
[1] 0.365
[1] 0.37
[1] 0.375
[1] 0.38
[1] 0.385
[1] 0.39
[1] 0.395
[1] 0.4
[1] 0.405
[1] 0.41
[1] 0.415
[1] 0.42
[1] 0.425
[1] 0.43
[1] 0.435
[1] 0.44
[1] 0.445
[1] 0.45
[1] 0.455
[1] 0.46
[1] 0.465
[1] 0.47
[1] 0.475
[1] 0.48
[1] 0.485
[1] 0.49
[1] 0.495
[1] 0.5
[1] 0.505
[1] 0.51
[1] 0.515
[1] 0.52
[1] 0.525
[1] 0.53
[1] 0.535
[1] 0.54
[1] 0.545
[1] 0.55
[1] 0.555
[1] 0.56
[1] 0.565
[1] 0.57
[1] 0.575
[1] 0.58
[1] 0.585
[1] 0.59
[1] 0.595
[1] 0.6
[1] 0.605
[1] 0.61
[1] 0.615
[1] 0.62
[1] 0.625
[1] 0.63
[1] 0.635
[1] 0.64
[1] 0.645
[1] 0.65
[1] 0.655
[1] 0.66
[1] 0.665
[1] 0.67
[1] 0.675
[1] 0.68
[1] 0.685
[1] 0.69
[1] 0.695
[1] 0.7
[1] 0.705
[1] 0.71
[1] 0.715
[1] 0.72
[1] 0.725
[1] 0.73
[1] 0.735
[1] 0.74
[1] 0.745
[1] 0.75
[1] 0.755
[1] 0.76
[1] 0.765
[1] 0.77
[1] 0.775
[1] 0.78
[1] 0.785
[1] 0.79
[1] 0.795
[1] 0.8
[1] 0.805
[1] 0.81
[1] 0.815
[1] 0.82
[1] 0.825
[1] 0.83
[1] 0.835
[1] 0.84
[1] 0.845
[1] 0.85
[1] 0.855
[1] 0.86
[1] 0.865
[1] 0.87
[1] 0.875
[1] 0.88
[1] 0.885
[1] 0.89
[1] 0.895
[1] 0.9
[1] 0.905
[1] 0.91
[1] 0.915
[1] 0.92
[1] 0.925
[1] 0.93
[1] 0.935
[1] 0.94
[1] 0.945
[1] 0.95
[1] 0.955
[1] 0.96
[1] 0.965
[1] 0.97
[1] 0.975
[1] 0.98
[1] 0.985
[1] 0.99
[1] 0.995
[1] 1

if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)

[1] 200

nrow(Y)

[1] 200

Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "A_dens", "trace")
(data <- cbind(input_var, Y)) %>% head()

Plot Results

pairs(data)

cor(data)

                     p        beta
p            1.0000000  0.00000000
beta         0.0000000  1.00000000
size         0.0000000  0.00000000
eigenvalue2 -0.3657134  0.57594018
A_dens       0.9944206 -0.01047333
trace       -0.3986436 -0.70810602
                   size eigenvalue2
p            0.00000000  -0.3657134
beta         0.00000000   0.5759402
size         1.00000000  -0.3528876
eigenvalue2 -0.35288757   1.0000000
A_dens       0.05079861  -0.4078047
trace       -0.04276000  -0.4008003
                 A_dens      trace
p            0.99442058 -0.3986436
beta        -0.01047333 -0.7081060
size         0.05079861 -0.0427600
eigenvalue2 -0.40780467 -0.4008003
A_dens       1.00000000 -0.3935818
trace       -0.39358178  1.0000000

library(corrplot)
cormat = cor(data, method = 'spearman')
corrplot(cormat, method = "ellipse", type = "lower")

names(data)

[1] "p"           "beta"        "size"       
[4] "eigenvalue2" "A_dens"      "trace"      

Let’s look at a 3d Output

# plot3d(data$beta, data$A_dens, data$eigenvalue2, size = 2)

names(data)

[1] "p"           "beta"        "size"       
[4] "eigenvalue2" "A_dens"      "trace"      

library(plotly)

d <- data
# d$beta <- log(d$beta)

fig <- plot_ly(d, x = ~A_dens, y = ~beta, z = ~eigenvalue2)
fig <- fig %>% add_markers(size = 1)
fig <- fig %>% layout(scene = list(xaxis = list(title = 'Density'),
                     yaxis = list(title = 'Beta'),
                     zaxis = list(title = 'E2')))

fig

NA

names(data)

[1] "p"           "beta"        "size"       
[4] "eigenvalue2" "A_dens"      "trace"      

library(plotly)

#d <- data[sample(1:nrow(data), 1000),]
d <- data

fig <- plot_ly(d, x = ~A_dens, y = ~trace, z = ~eigenvalue2)
fig <- fig %>% add_markers(size = 1)
fig <- fig %>% layout(scene = list(xaxis = list(title = 'Density'),
                     yaxis = list(title = 'trace'),
                     zaxis = list(title = 'E2')))

fig

NA

Clearly I should be able to model this.

Let’s look at density, it’s continuous so I should be able to take splices of A_dens.

A_dens is dependent on p, so I’ll just use different ps

p    <- seq(from = 0.1, to = 0.95, length.out = 5)
beta <- seq(from = 1, to = 10, length.out = 1000)
size = 100
input_var <- expand.grid("p" = p, "beta" = beta, "size" = size)
input_var

nc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$p, X$beta, X$size)
  print(i/nrow(input_var))
}

[1] 2e-04
[1] 4e-04
[1] 6e-04
[1] 8e-04
[1] 0.001
[1] 0.0012
[1] 0.0014
[1] 0.0016
[1] 0.0018
[1] 0.002
[1] 0.0022
[1] 0.0024
[1] 0.0026
[1] 0.0028
[1] 0.003
[1] 0.0032
[1] 0.0034
[1] 0.0036
[1] 0.0038
[1] 0.004
[1] 0.0042
[1] 0.0044
[1] 0.0046
[1] 0.0048
[1] 0.005
[1] 0.0052
[1] 0.0054
[1] 0.0056
[1] 0.0058
[1] 0.006
[1] 0.0062
[1] 0.0064
[1] 0.0066
[1] 0.0068
[1] 0.007
[1] 0.0072
[1] 0.0074
[1] 0.0076
[1] 0.0078
[1] 0.008
[1] 0.0082
[1] 0.0084
[1] 0.0086
[1] 0.0088
[1] 0.009
[1] 0.0092
[1] 0.0094
[1] 0.0096
[1] 0.0098
[1] 0.01
[1] 0.0102
[1] 0.0104
[1] 0.0106
[1] 0.0108
[1] 0.011
[1] 0.0112
[1] 0.0114
[1] 0.0116
[1] 0.0118
[1] 0.012
[1] 0.0122
[1] 0.0124
[1] 0.0126
[1] 0.0128
[1] 0.013
[1] 0.0132
[1] 0.0134
[1] 0.0136
[1] 0.0138
[1] 0.014
[1] 0.0142
[1] 0.0144
[1] 0.0146
[1] 0.0148
[1] 0.015
[1] 0.0152
[1] 0.0154
[1] 0.0156
[1] 0.0158
[1] 0.016
[1] 0.0162
[1] 0.0164
[1] 0.0166
[1] 0.0168
[1] 0.017
[1] 0.0172
[1] 0.0174
[1] 0.0176
[1] 0.0178
[1] 0.018
[1] 0.0182
[1] 0.0184
[1] 0.0186
[1] 0.0188
[1] 0.019
[1] 0.0192
[1] 0.0194
[1] 0.0196
[1] 0.0198
[1] 0.02
[1] 0.0202
[1] 0.0204
[1] 0.0206
[1] 0.0208
[1] 0.021
[1] 0.0212
[1] 0.0214
[1] 0.0216
[1] 0.0218
[1] 0.022
[1] 0.0222
[1] 0.0224
[1] 0.0226
[1] 0.0228
[1] 0.023
[1] 0.0232
[1] 0.0234
[1] 0.0236
[1] 0.0238
[1] 0.024
[1] 0.0242
[1] 0.0244
[1] 0.0246
[1] 0.0248
[1] 0.025
[1] 0.0252
[1] 0.0254
[1] 0.0256
[1] 0.0258
[1] 0.026
[1] 0.0262
[1] 0.0264
[1] 0.0266
[1] 0.0268
[1] 0.027
[1] 0.0272
[1] 0.0274
[1] 0.0276
[1] 0.0278
[1] 0.028
[1] 0.0282
[1] 0.0284
[1] 0.0286
[1] 0.0288
[1] 0.029
[1] 0.0292
[1] 0.0294
[1] 0.0296
[1] 0.0298
[1] 0.03
[1] 0.0302
[1] 0.0304
[1] 0.0306
[1] 0.0308
[1] 0.031
[1] 0.0312
[1] 0.0314
[1] 0.0316
[1] 0.0318
[1] 0.032
[1] 0.0322
[1] 0.0324
[1] 0.0326
[1] 0.0328
[1] 0.033
[1] 0.0332
[1] 0.0334
[1] 0.0336
[1] 0.0338
[1] 0.034
[1] 0.0342
[1] 0.0344
[1] 0.0346
[1] 0.0348
[1] 0.035
[1] 0.0352
[1] 0.0354
[1] 0.0356
[1] 0.0358
[1] 0.036
[1] 0.0362
[1] 0.0364
[1] 0.0366
[1] 0.0368
[1] 0.037
[1] 0.0372
[1] 0.0374
[1] 0.0376
[1] 0.0378
[1] 0.038
[1] 0.0382
[1] 0.0384
[1] 0.0386
[1] 0.0388
[1] 0.039
[1] 0.0392
[1] 0.0394
[1] 0.0396
[1] 0.0398
[1] 0.04
[1] 0.0402
[1] 0.0404
[1] 0.0406
[1] 0.0408
[1] 0.041
[1] 0.0412
[1] 0.0414
[1] 0.0416
[1] 0.0418
[1] 0.042
[1] 0.0422
[1] 0.0424
[1] 0.0426
[1] 0.0428
[1] 0.043
[1] 0.0432
[1] 0.0434
[1] 0.0436
[1] 0.0438
[1] 0.044
[1] 0.0442
[1] 0.0444
[1] 0.0446
[1] 0.0448
[1] 0.045
[1] 0.0452
[1] 0.0454
[1] 0.0456
[1] 0.0458
[1] 0.046
[1] 0.0462
[1] 0.0464
[1] 0.0466
[1] 0.0468
[1] 0.047
[1] 0.0472
[1] 0.0474
[1] 0.0476
[1] 0.0478
[1] 0.048
[1] 0.0482
[1] 0.0484
[1] 0.0486
[1] 0.0488
[1] 0.049
[1] 0.0492
[1] 0.0494
[1] 0.0496
[1] 0.0498
[1] 0.05
[1] 0.0502
[1] 0.0504
[1] 0.0506
[1] 0.0508
[1] 0.051
[1] 0.0512
[1] 0.0514
[1] 0.0516
[1] 0.0518
[1] 0.052
[1] 0.0522
[1] 0.0524
[1] 0.0526
[1] 0.0528
[1] 0.053
[1] 0.0532
[1] 0.0534
[1] 0.0536
[1] 0.0538
[1] 0.054
[1] 0.0542
[1] 0.0544
[1] 0.0546
[1] 0.0548
[1] 0.055
[1] 0.0552
[1] 0.0554
[1] 0.0556
[1] 0.0558
[1] 0.056
[1] 0.0562
[1] 0.0564
[1] 0.0566
[1] 0.0568
[1] 0.057
[1] 0.0572
[1] 0.0574
[1] 0.0576
[1] 0.0578
[1] 0.058
[1] 0.0582
[1] 0.0584
[1] 0.0586
[1] 0.0588
[1] 0.059
[1] 0.0592
[1] 0.0594
[1] 0.0596
[1] 0.0598
[1] 0.06
[1] 0.0602
[1] 0.0604
[1] 0.0606
[1] 0.0608
[1] 0.061
[1] 0.0612
[1] 0.0614
[1] 0.0616
[1] 0.0618
[1] 0.062
[1] 0.0622
[1] 0.0624
[1] 0.0626
[1] 0.0628
[1] 0.063
[1] 0.0632
[1] 0.0634
[1] 0.0636
[1] 0.0638
[1] 0.064
[1] 0.0642
[1] 0.0644
[1] 0.0646
[1] 0.0648
[1] 0.065
[1] 0.0652
[1] 0.0654
[1] 0.0656
[1] 0.0658
[1] 0.066
[1] 0.0662
[1] 0.0664
[1] 0.0666
[1] 0.0668
[1] 0.067
[1] 0.0672
[1] 0.0674
[1] 0.0676
[1] 0.0678
[1] 0.068
[1] 0.0682
[1] 0.0684
[1] 0.0686
[1] 0.0688
[1] 0.069
[1] 0.0692
[1] 0.0694
[1] 0.0696
[1] 0.0698
[1] 0.07
[1] 0.0702
[1] 0.0704
[1] 0.0706
[1] 0.0708
[1] 0.071
[1] 0.0712
[1] 0.0714
[1] 0.0716
[1] 0.0718
[1] 0.072
[1] 0.0722
[1] 0.0724
[1] 0.0726
[1] 0.0728
[1] 0.073
[1] 0.0732
[1] 0.0734
[1] 0.0736
[1] 0.0738
[1] 0.074
[1] 0.0742
[1] 0.0744
[1] 0.0746
[1] 0.0748
[1] 0.075
[1] 0.0752
[1] 0.0754
[1] 0.0756
[1] 0.0758
[1] 0.076
[1] 0.0762
[1] 0.0764
[1] 0.0766
[1] 0.0768
[1] 0.077
[1] 0.0772
[1] 0.0774
[1] 0.0776
[1] 0.0778
[1] 0.078
[1] 0.0782
[1] 0.0784
[1] 0.0786
[1] 0.0788
[1] 0.079
[1] 0.0792
[1] 0.0794
[1] 0.0796
[1] 0.0798
[1] 0.08
[1] 0.0802
[1] 0.0804
[1] 0.0806
[1] 0.0808
[1] 0.081
[1] 0.0812
[1] 0.0814
[1] 0.0816
[1] 0.0818
[1] 0.082
[1] 0.0822
[1] 0.0824
[1] 0.0826
[1] 0.0828
[1] 0.083
[1] 0.0832
[1] 0.0834
[1] 0.0836
[1] 0.0838
[1] 0.084
[1] 0.0842
[1] 0.0844
[1] 0.0846
[1] 0.0848
[1] 0.085
[1] 0.0852
[1] 0.0854
[1] 0.0856
[1] 0.0858
[1] 0.086
[1] 0.0862
[1] 0.0864
[1] 0.0866
[1] 0.0868
[1] 0.087
[1] 0.0872
[1] 0.0874
[1] 0.0876
[1] 0.0878
[1] 0.088
[1] 0.0882
[1] 0.0884
[1] 0.0886
[1] 0.0888
[1] 0.089
[1] 0.0892
[1] 0.0894
[1] 0.0896
[1] 0.0898
[1] 0.09
[1] 0.0902
[1] 0.0904
[1] 0.0906
[1] 0.0908
[1] 0.091
[1] 0.0912
[1] 0.0914
[1] 0.0916
[1] 0.0918
[1] 0.092
[1] 0.0922
[1] 0.0924
[1] 0.0926
[1] 0.0928
[1] 0.093
[1] 0.0932
[1] 0.0934
[1] 0.0936
[1] 0.0938
[1] 0.094
[1] 0.0942
[1] 0.0944
[1] 0.0946
[1] 0.0948
[1] 0.095
[1] 0.0952
[1] 0.0954
[1] 0.0956
[1] 0.0958
[1] 0.096
[1] 0.0962
[1] 0.0964
[1] 0.0966
[1] 0.0968
[1] 0.097
[1] 0.0972
[1] 0.0974
[1] 0.0976
[1] 0.0978
[1] 0.098
[1] 0.0982
[1] 0.0984
[1] 0.0986
[1] 0.0988
[1] 0.099
[1] 0.0992
[1] 0.0994
[1] 0.0996
[1] 0.0998
[1] 0.1
[1] 0.1002
[1] 0.1004
[1] 0.1006
[1] 0.1008
[1] 0.101
[1] 0.1012
[1] 0.1014
[1] 0.1016
[1] 0.1018
[1] 0.102
[1] 0.1022
[1] 0.1024
[1] 0.1026
[1] 0.1028
[1] 0.103
[1] 0.1032
[1] 0.1034
[1] 0.1036
[1] 0.1038
[1] 0.104
[1] 0.1042
[1] 0.1044
[1] 0.1046
[1] 0.1048
[1] 0.105
[1] 0.1052
[1] 0.1054
[1] 0.1056
[1] 0.1058
[1] 0.106
[1] 0.1062
[1] 0.1064
[1] 0.1066
[1] 0.1068
[1] 0.107
[1] 0.1072
[1] 0.1074
[1] 0.1076
[1] 0.1078
[1] 0.108
[1] 0.1082
[1] 0.1084
[1] 0.1086
[1] 0.1088
[1] 0.109
[1] 0.1092
[1] 0.1094
[1] 0.1096
[1] 0.1098
[1] 0.11
[1] 0.1102
[1] 0.1104
[1] 0.1106
[1] 0.1108
[1] 0.111
[1] 0.1112
[1] 0.1114
[1] 0.1116
[1] 0.1118
[1] 0.112
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[1] 0.854
[1] 0.8542
[1] 0.8544
[1] 0.8546
[1] 0.8548
[1] 0.855
[1] 0.8552
[1] 0.8554
[1] 0.8556
[1] 0.8558
[1] 0.856
[1] 0.8562
[1] 0.8564
[1] 0.8566
[1] 0.8568
[1] 0.857
[1] 0.8572
[1] 0.8574
[1] 0.8576
[1] 0.8578
[1] 0.858
[1] 0.8582
[1] 0.8584
[1] 0.8586
[1] 0.8588
[1] 0.859
[1] 0.8592
[1] 0.8594
[1] 0.8596
[1] 0.8598
[1] 0.86
[1] 0.8602
[1] 0.8604
[1] 0.8606
[1] 0.8608
[1] 0.861
[1] 0.8612
[1] 0.8614
[1] 0.8616
[1] 0.8618
[1] 0.862
[1] 0.8622
[1] 0.8624
[1] 0.8626
[1] 0.8628
[1] 0.863
[1] 0.8632
[1] 0.8634
[1] 0.8636
[1] 0.8638
[1] 0.864
[1] 0.8642
[1] 0.8644
[1] 0.8646
[1] 0.8648
[1] 0.865
[1] 0.8652
[1] 0.8654
[1] 0.8656
[1] 0.8658
[1] 0.866
[1] 0.8662
[1] 0.8664
[1] 0.8666
[1] 0.8668
[1] 0.867
[1] 0.8672
[1] 0.8674
[1] 0.8676
[1] 0.8678
[1] 0.868
[1] 0.8682
[1] 0.8684
[1] 0.8686
[1] 0.8688
[1] 0.869
[1] 0.8692
[1] 0.8694
[1] 0.8696
[1] 0.8698
[1] 0.87
[1] 0.8702
[1] 0.8704
[1] 0.8706
[1] 0.8708
[1] 0.871
[1] 0.8712
[1] 0.8714
[1] 0.8716
[1] 0.8718
[1] 0.872
[1] 0.8722
[1] 0.8724
[1] 0.8726
[1] 0.8728
[1] 0.873
[1] 0.8732
[1] 0.8734
[1] 0.8736
[1] 0.8738
[1] 0.874
[1] 0.8742
[1] 0.8744
[1] 0.8746
[1] 0.8748
[1] 0.875
[1] 0.8752
[1] 0.8754
[1] 0.8756
[1] 0.8758
[1] 0.876
[1] 0.8762
[1] 0.8764
[1] 0.8766
[1] 0.8768
[1] 0.877
[1] 0.8772
[1] 0.8774
[1] 0.8776
[1] 0.8778
[1] 0.878
[1] 0.8782
[1] 0.8784
[1] 0.8786
[1] 0.8788
[1] 0.879
[1] 0.8792
[1] 0.8794
[1] 0.8796
[1] 0.8798
[1] 0.88
[1] 0.8802
[1] 0.8804
[1] 0.8806
[1] 0.8808
[1] 0.881
[1] 0.8812
[1] 0.8814
[1] 0.8816
[1] 0.8818
[1] 0.882
[1] 0.8822
[1] 0.8824
[1] 0.8826
[1] 0.8828
[1] 0.883
[1] 0.8832
[1] 0.8834
[1] 0.8836
[1] 0.8838
[1] 0.884
[1] 0.8842
[1] 0.8844
[1] 0.8846
[1] 0.8848
[1] 0.885
[1] 0.8852
[1] 0.8854
[1] 0.8856
[1] 0.8858
[1] 0.886
[1] 0.8862
[1] 0.8864
[1] 0.8866
[1] 0.8868
[1] 0.887
[1] 0.8872
[1] 0.8874
[1] 0.8876
[1] 0.8878
[1] 0.888
[1] 0.8882
[1] 0.8884
[1] 0.8886
[1] 0.8888
[1] 0.889
[1] 0.8892
[1] 0.8894
[1] 0.8896
[1] 0.8898
[1] 0.89
[1] 0.8902
[1] 0.8904
[1] 0.8906
[1] 0.8908
[1] 0.891
[1] 0.8912
[1] 0.8914
[1] 0.8916
[1] 0.8918
[1] 0.892
[1] 0.8922
[1] 0.8924
[1] 0.8926
[1] 0.8928
[1] 0.893
[1] 0.8932
[1] 0.8934
[1] 0.8936
[1] 0.8938
[1] 0.894
[1] 0.8942
[1] 0.8944
[1] 0.8946
[1] 0.8948
[1] 0.895
[1] 0.8952
[1] 0.8954
[1] 0.8956
[1] 0.8958
[1] 0.896
[1] 0.8962
[1] 0.8964
[1] 0.8966
[1] 0.8968
[1] 0.897
[1] 0.8972
[1] 0.8974
[1] 0.8976
[1] 0.8978
[1] 0.898
[1] 0.8982
[1] 0.8984
[1] 0.8986
[1] 0.8988
[1] 0.899
[1] 0.8992
[1] 0.8994
[1] 0.8996
[1] 0.8998
[1] 0.9
[1] 0.9002
[1] 0.9004
[1] 0.9006
[1] 0.9008
[1] 0.901
[1] 0.9012
[1] 0.9014
[1] 0.9016
[1] 0.9018
[1] 0.902
[1] 0.9022
[1] 0.9024
[1] 0.9026
[1] 0.9028
[1] 0.903
[1] 0.9032
[1] 0.9034
[1] 0.9036
[1] 0.9038
[1] 0.904
[1] 0.9042
[1] 0.9044
[1] 0.9046
[1] 0.9048
[1] 0.905
[1] 0.9052
[1] 0.9054
[1] 0.9056
[1] 0.9058
[1] 0.906
[1] 0.9062
[1] 0.9064
[1] 0.9066
[1] 0.9068
[1] 0.907
[1] 0.9072
[1] 0.9074
[1] 0.9076
[1] 0.9078
[1] 0.908
[1] 0.9082
[1] 0.9084
[1] 0.9086
[1] 0.9088
[1] 0.909
[1] 0.9092
[1] 0.9094
[1] 0.9096
[1] 0.9098
[1] 0.91
[1] 0.9102
[1] 0.9104
[1] 0.9106
[1] 0.9108
[1] 0.911
[1] 0.9112
[1] 0.9114
[1] 0.9116
[1] 0.9118
[1] 0.912
[1] 0.9122
[1] 0.9124
[1] 0.9126
[1] 0.9128
[1] 0.913
[1] 0.9132
[1] 0.9134
[1] 0.9136
[1] 0.9138
[1] 0.914
[1] 0.9142
[1] 0.9144
[1] 0.9146
[1] 0.9148
[1] 0.915
[1] 0.9152
[1] 0.9154
[1] 0.9156
[1] 0.9158
[1] 0.916
[1] 0.9162
[1] 0.9164
[1] 0.9166
[1] 0.9168
[1] 0.917
[1] 0.9172
[1] 0.9174
[1] 0.9176
[1] 0.9178
[1] 0.918
[1] 0.9182
[1] 0.9184
[1] 0.9186
[1] 0.9188
[1] 0.919
[1] 0.9192
[1] 0.9194
[1] 0.9196
[1] 0.9198
[1] 0.92
[1] 0.9202
[1] 0.9204
[1] 0.9206
[1] 0.9208
[1] 0.921
[1] 0.9212
[1] 0.9214
[1] 0.9216
[1] 0.9218
[1] 0.922
[1] 0.9222
[1] 0.9224
[1] 0.9226
[1] 0.9228
[1] 0.923
[1] 0.9232
[1] 0.9234
[1] 0.9236
[1] 0.9238
[1] 0.924
[1] 0.9242
[1] 0.9244
[1] 0.9246
[1] 0.9248
[1] 0.925
[1] 0.9252
[1] 0.9254
[1] 0.9256
[1] 0.9258
[1] 0.926
[1] 0.9262
[1] 0.9264
[1] 0.9266
[1] 0.9268
[1] 0.927
[1] 0.9272
[1] 0.9274
[1] 0.9276
[1] 0.9278
[1] 0.928
[1] 0.9282
[1] 0.9284
[1] 0.9286
[1] 0.9288
[1] 0.929
[1] 0.9292
[1] 0.9294
[1] 0.9296
[1] 0.9298
[1] 0.93
[1] 0.9302
[1] 0.9304
[1] 0.9306
[1] 0.9308
[1] 0.931
[1] 0.9312
[1] 0.9314
[1] 0.9316
[1] 0.9318
[1] 0.932
[1] 0.9322
[1] 0.9324
[1] 0.9326
[1] 0.9328
[1] 0.933
[1] 0.9332
[1] 0.9334
[1] 0.9336
[1] 0.9338
[1] 0.934
[1] 0.9342
[1] 0.9344
[1] 0.9346
[1] 0.9348
[1] 0.935
[1] 0.9352
[1] 0.9354
[1] 0.9356
[1] 0.9358
[1] 0.936
[1] 0.9362
[1] 0.9364
[1] 0.9366
[1] 0.9368
[1] 0.937
[1] 0.9372
[1] 0.9374
[1] 0.9376
[1] 0.9378
[1] 0.938
[1] 0.9382
[1] 0.9384
[1] 0.9386
[1] 0.9388
[1] 0.939
[1] 0.9392
[1] 0.9394
[1] 0.9396
[1] 0.9398
[1] 0.94
[1] 0.9402
[1] 0.9404
[1] 0.9406
[1] 0.9408
[1] 0.941
[1] 0.9412
[1] 0.9414
[1] 0.9416
[1] 0.9418
[1] 0.942
[1] 0.9422
[1] 0.9424
[1] 0.9426
[1] 0.9428
[1] 0.943
[1] 0.9432
[1] 0.9434
[1] 0.9436
[1] 0.9438
[1] 0.944
[1] 0.9442
[1] 0.9444
[1] 0.9446
[1] 0.9448
[1] 0.945
[1] 0.9452
[1] 0.9454
[1] 0.9456
[1] 0.9458
[1] 0.946
[1] 0.9462
[1] 0.9464
[1] 0.9466
[1] 0.9468
[1] 0.947
[1] 0.9472
[1] 0.9474
[1] 0.9476
[1] 0.9478
[1] 0.948
[1] 0.9482
[1] 0.9484
[1] 0.9486
[1] 0.9488
[1] 0.949
[1] 0.9492
[1] 0.9494
[1] 0.9496
[1] 0.9498
[1] 0.95
[1] 0.9502
[1] 0.9504
[1] 0.9506
[1] 0.9508
[1] 0.951
[1] 0.9512
[1] 0.9514
[1] 0.9516
[1] 0.9518
[1] 0.952
[1] 0.9522
[1] 0.9524
[1] 0.9526
[1] 0.9528
[1] 0.953
[1] 0.9532
[1] 0.9534
[1] 0.9536
[1] 0.9538
[1] 0.954
[1] 0.9542
[1] 0.9544
[1] 0.9546
[1] 0.9548
[1] 0.955
[1] 0.9552
[1] 0.9554
[1] 0.9556
[1] 0.9558
[1] 0.956
[1] 0.9562
[1] 0.9564
[1] 0.9566
[1] 0.9568
[1] 0.957
[1] 0.9572
[1] 0.9574
[1] 0.9576
[1] 0.9578
[1] 0.958
[1] 0.9582
[1] 0.9584
[1] 0.9586
[1] 0.9588
[1] 0.959
[1] 0.9592
[1] 0.9594
[1] 0.9596
[1] 0.9598
[1] 0.96
[1] 0.9602
[1] 0.9604
[1] 0.9606
[1] 0.9608
[1] 0.961
[1] 0.9612
[1] 0.9614
[1] 0.9616
[1] 0.9618
[1] 0.962
[1] 0.9622
[1] 0.9624
[1] 0.9626
[1] 0.9628
[1] 0.963
[1] 0.9632
[1] 0.9634
[1] 0.9636
[1] 0.9638
[1] 0.964
[1] 0.9642
[1] 0.9644
[1] 0.9646
[1] 0.9648
[1] 0.965
[1] 0.9652
[1] 0.9654
[1] 0.9656
[1] 0.9658
[1] 0.966
[1] 0.9662
[1] 0.9664
[1] 0.9666
[1] 0.9668
[1] 0.967
[1] 0.9672
[1] 0.9674
[1] 0.9676
[1] 0.9678
[1] 0.968
[1] 0.9682
[1] 0.9684
[1] 0.9686
[1] 0.9688
[1] 0.969
[1] 0.9692
[1] 0.9694
[1] 0.9696
[1] 0.9698
[1] 0.97
[1] 0.9702
[1] 0.9704
[1] 0.9706
[1] 0.9708
[1] 0.971
[1] 0.9712
[1] 0.9714
[1] 0.9716
[1] 0.9718
[1] 0.972
[1] 0.9722
[1] 0.9724
[1] 0.9726
[1] 0.9728
[1] 0.973
[1] 0.9732
[1] 0.9734
[1] 0.9736
[1] 0.9738
[1] 0.974
[1] 0.9742
[1] 0.9744
[1] 0.9746
[1] 0.9748
[1] 0.975
[1] 0.9752
[1] 0.9754
[1] 0.9756
[1] 0.9758
[1] 0.976
[1] 0.9762
[1] 0.9764
[1] 0.9766
[1] 0.9768
[1] 0.977
[1] 0.9772
[1] 0.9774
[1] 0.9776
[1] 0.9778
[1] 0.978
[1] 0.9782
[1] 0.9784
[1] 0.9786
[1] 0.9788
[1] 0.979
[1] 0.9792
[1] 0.9794
[1] 0.9796
[1] 0.9798
[1] 0.98
[1] 0.9802
[1] 0.9804
[1] 0.9806
[1] 0.9808
[1] 0.981
[1] 0.9812
[1] 0.9814
[1] 0.9816
[1] 0.9818
[1] 0.982
[1] 0.9822
[1] 0.9824
[1] 0.9826
[1] 0.9828
[1] 0.983
[1] 0.9832
[1] 0.9834
[1] 0.9836
[1] 0.9838
[1] 0.984
[1] 0.9842
[1] 0.9844
[1] 0.9846
[1] 0.9848
[1] 0.985
[1] 0.9852
[1] 0.9854
[1] 0.9856
[1] 0.9858
[1] 0.986
[1] 0.9862
[1] 0.9864
[1] 0.9866
[1] 0.9868
[1] 0.987
[1] 0.9872
[1] 0.9874
[1] 0.9876
[1] 0.9878
[1] 0.988
[1] 0.9882
[1] 0.9884
[1] 0.9886
[1] 0.9888
[1] 0.989
[1] 0.9892
[1] 0.9894
[1] 0.9896
[1] 0.9898
[1] 0.99
[1] 0.9902
[1] 0.9904
[1] 0.9906
[1] 0.9908
[1] 0.991
[1] 0.9912
[1] 0.9914
[1] 0.9916
[1] 0.9918
[1] 0.992
[1] 0.9922
[1] 0.9924
[1] 0.9926
[1] 0.9928
[1] 0.993
[1] 0.9932
[1] 0.9934
[1] 0.9936
[1] 0.9938
[1] 0.994
[1] 0.9942
[1] 0.9944
[1] 0.9946
[1] 0.9948
[1] 0.995
[1] 0.9952
[1] 0.9954
[1] 0.9956
[1] 0.9958
[1] 0.996
[1] 0.9962
[1] 0.9964
[1] 0.9966
[1] 0.9968
[1] 0.997
[1] 0.9972
[1] 0.9974
[1] 0.9976
[1] 0.9978
[1] 0.998
[1] 0.9982
[1] 0.9984
[1] 0.9986
[1] 0.9988
[1] 0.999
[1] 0.9992
[1] 0.9994
[1] 0.9996
[1] 0.9998
[1] 1

if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)

[1] 5000

nrow(Y)

[1] 5000

Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "trace")
(data2 <- cbind(input_var, Y)) %>% head()

data2$p <- factor(data2$p)

ggplot(data2[30:nrow(data2),], mapping = aes(col = p, x = beta, y = eigenvalue2)) +
  geom_point(size = 0.5) +
  stat_smooth() +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "Beta", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
  guides(col = guide_legend("Link Density"))  +
  theme_bw()

These look logarithmic, let’s do a log transform:

ggplot(data2[30:nrow(data2),], mapping = aes(col = p, x = log(beta), y = eigenvalue2)) +
  geom_point(size = 0.5) +
  stat_smooth() +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "Beta", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
  guides(col = guide_legend("Link Density"))  +
  theme_bw()

Link density is a function of size so E2ln(Beta) is still pretty useful.

it seems that this is very nearly linear for low densities, the web would have a low density, for example ba graphs appear bounded above by 1/2:

n_vec <- 0:20
m <- 3
y <- c()
for (n in n_vec) {
    x <- sample_pa(n = n, power = 3, m = m) %>% 
      get.adjacency() %>% 
      mean()
    y <- c(y, x)
}
plot(n_vec, y)

length(n_vec)

[1] 21

length(y)

[1] 21

So considering small Densities is wise, let’s have a look.

So at this stage I would like to fit a model over what we have in the first plotly diagram, but, I’m going to look at ba graphs in case there is something more insightful.

ggplot(data2[30:nrow(data2),], mapping = aes(col = factor(p), x = log(beta*p), y = eigenvalue2)) +
  geom_point(size = 0.5) +
  stat_smooth() +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "Beta", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
  
  guides(col = guide_legend("Link Density"))  +
  theme_bw()

It seems for very low densities that it is linear and then logarithmic for moderate densities. BA models have density~=m/n so for the BA model we’ll just use beta.

p       <- seq(from = 0.1, to = 0.95, length.out = 40)
size    <- seq(from = 100, to = 1000, length.out = 5)
beta <- seq(from = 1, to = 10, length.out = 1000)
beta <- 5
input_var <- expand.grid("p" = p, "beta" = beta, "size" = size)
input_var

nc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$p, X$beta, X$size)
  print(i/nrow(input_var))
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)
nrow(Y)
Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "trace")
(data2 <- cbind(input_var, Y)) %>% head()
data2$p <- factor(data2$p)



ggplot(data2, mapping = aes(col = factor(p), x = log(size), y = eigenvalue2)) +
  geom_point(size = 0.5) +
#  stat_smooth() +
#  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "P", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
  guides(col = guide_legend("Size"))  +
  theme_bw()